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I am trying to find out some facts about the group SO(2,1). Specifically; Is the exponential map onto? If so, can the Haar measure be written in terms of the Lebesgue integral over a suitable subset of the Lie algebra? What is that subset?

If the exponential map isn't onto, then for what subset of the Lie algebra is it at least 1-1 (injective)?

I was hoping there would be a reference out there that explained these things, but have so far been unable to find it.

Thank you.

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# SO(2,1) - Haar measure, exponential map

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